Maximizing Likelihood Functions Using JMP's Nonlinear Fit Platform

Class Objectives:

• Learn to use JMP's Nonlinear platform to calculate maximum likelihood estimates in the single sample case directly from the likelihood function.

Homework Assignments:

• Use the Nonlinear Fit platform of JMP to obtain the MLEs for mu and sigma of a lognormal model applied to the:
• Data on the number of revolutions to failure for each of 23 ball bearings. (Example 2.1, Page 37 of your textbook.)
• Data on the strengths of 48 pieces of weathered braided cord. (Example 2.3, Page 46 of your textbook.)

Recall that you can download the JMP files for these data sets if you go to the end of this class summary.

This homework is due on October 23.

Class Outline and Main Points:

• JMP's Nonlinear Fit platform
• Object function
• Model function
• Loss function
• Weibull and Gumbel case
• Using JMP to obtain MLEs and their variance-covariance matrix
• The negative of log-likelihood is used as the object function in JMP's Nonlinear Fit platform
• Case I: Data in not censored
• Case II: Data is right censored
• Using JMP to verify that the Nonlinear Fit platform obtained the right values for the MLEs

Maximizing Likelihood Functions Using JMP's Nonlinear Fit Platform

We illustrate how JMP calculates maximum likelihood estimates using the Ball Bearing data of Example 2.1, Page 37 of your textbook. In particular, we obtain the MLEs for the parameters of a Gumbel model applied to the log data. From these MLE values we can obtain the MLEs for the parameters of the Weibull model of the original data. This follows since the Weibull scale parameter is log(mu) and the Weibull shape parameter is 1/sigma where mu and sigma are the location and scale parameters of the Gumbel model of the log data.

Here are the steps one must go through with JMP:

• Derive a new column, Gumbel loss, using the calculator. The first step is to use the Parameters function of the calculator to define two parameters mu and sigma. Assign initial values to these parameters as shown below. We use the graphical estimates based on the Q-Q plot for these initial values. Getting "good" initial values is very important whenever one is using recursive numerical optimization techniques.

• Finish defining the column Gumbel loss using the following formula programed into the calculator.

• Select the Nonlinear Fit platform from the Analyze menu:

• Assign the Gumbel loss column to the Loss slot in the resulting pop-up menu and click OK. (We can do this because the loss function is the identity function.)

• Check the Loss is -LogLikelihood option in the resulting pop-up model and click Reset and Go. Wait until the iterations stop and click on Confid Limits. Open all closed output boxes. Below are the results:

Notice that JMP arrived at the maximum likelihood estimates given in the book. It also calculated the equivalent of the variance-covariance matrix, namely, the standard errors and the correlation of the estimates. In addition, it calculated the likelihood ratio confidence intervals. As we have discussed these are usually better than the confidence intervals based on standard errors.

Verifying Optimization Results Obtained Using JMP's Nonlinear Fit Platform

The results of an iterative numerical optimization technique should be verified. One could have converged to a local minimum or an inflection point rather than to the global minimum.

We illustrate how JMP can be used to verify the MLEs obtained using the Nonlinear Fit platform. We continue with the ball bearing example at the point of the last JMP screen shot shown above.

Here are the steps one must go through with JMP:

• Go to the \$ (dollar) menu and select as follows:

• The following output will result. By clicking go we will generate an eleven by eleven grid of points centered on the MLEs for mu and sigma. The default Mins and Maxs shown below are 2.5 Standard Errors below and above the MLEs. You can change these default values if you so choose.

• Below is a JMP table with the resulting grid. The quantity SSE is the negative of the log-likelihood for the values of mu and sigma given in the first two columns.

• Now go to JMP's Graph menu and select as follows:

• Complete the resulting pop-up menu as shown below:

• When you click OK we get the following contour plot:

• As we can see the negative log-likelihood (SSE) does seem to have its minimum value at the MLE values arrived at by JMP (and the textbook), namely,

mu = 4.40
sigma = 0.4758

(Notice that we have only checked a neighborhood around the MLE values computed by JMP. Hence, we cannot be completely sure that these values are really the one that minimize the negative of the log-likelihood function. Nevertheless, we are very happy with the results of the contour plot!)

A fancier - though less useful - way to show that JMP's Nonlinear platform arrived at the true minimum of the negative log-likelihood function is to use JMP's Spinning Plot:

Below are the steps one must go through with JMP:

This spinning plot is very impressive when shown in a computer screen or projector.

Maximizing Likelihood Functions Using JMP's Nonlinear Fit Platform When the Data is Right Censored.

We use the data on the strengths of 48 pieces of weathered braided cord (Example 2.3, Page 46 of your textbook ) to illustrate the steps that one needs to go through to obtain MLEs in the right censored case.

The following screen shots should make it clear how to modify the steps provided above to accommodate right censoring.

As one can see here we also obtain the MLEs for the Weibull model of the cord data. Notice that the values coincide with those provided in the textbook.

Study Questions

• Why do you need to know how to calculate MLEs directly from the likelihood function? Why not relay on canned packages?
• Why do you need to know something about optimization to be able to use JMP's Nonlinear Fit platform effectively?
• What crucial information is missing when one uses JMP's Survival platform to obtain MLEs for Gumbel, Weibull, and lognormal models?

SAS JMP files (Mac) of classroom examples and homework

• Data on the number of revolutions to failure for each of 23 ball bearings. (Example 2.1, Page 37 of your textbook)

• Data on the strengths of 48 pieces of weathered braided cord. (Example 2.3, Page 46 of your textbook)